{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a finite [[bipartite graph|07DVHJ8N0A]] with parts $X$ and $Y$.\nFor $W \\subseteq X$, let $N(W)$ be the set of vertices of $Y$ that are adjacent to some vertex of $W$.\nThen $G$ has a [[matching|0Z87BHAFTB]] that covers every vertex of $X$ if and only if $|N(W)| \\ge |W|$ for every $W \\subseteq X$.\n\n## Remarks\n\nIf $|X| = |Y|$, such a matching is a [[perfect matching|05ZPH8KF9K]]. Thus the condition then characterizes the perfect matchings.\nA consequence: a finite regular bipartite graph with at least one edge has a perfect matching.\nThe theorem is the graph form of [[Hall's marriage theorem|08WCB90WND]]: let $X$ index the sets and let $Y$ be their union.\n\n## Explanations\n\n### L1 human\n\nThink of $X$ as workers and $Y$ as jobs, with an edge when a worker can do a job.\nEvery worker can get a different job exactly when this holds.\nEach group of workers can do, together, at least as many jobs as there are workers in the group.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "05C70"
    ],
    "name": "Hall's theorem for bipartite graphs",
    "role": "theorem",
    "sources": [
      {
        "locator": "Graph theoretic formulation",
        "url": "https://en.wikipedia.org/wiki/Hall's_marriage_theorem"
      }
    ],
    "tag": "0X2GJPNPFN",
    "uses": [
      "01T3TMGTTE",
      "07DVHJ8N0A",
      "0Z87BHAFTB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0X2GJPNPFN/",
  "links": [
    {
      "field": "uses",
      "target": "01T3TMGTTE"
    },
    {
      "field": "uses",
      "target": "07DVHJ8N0A"
    },
    {
      "field": "uses",
      "target": "0Z87BHAFTB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0X2GJPNPFN.md",
  "msc": [
    "05C70"
  ],
  "name": "Hall's theorem for bipartite graphs",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0X2/0X2GJPNPFN.md",
  "status": null,
  "tag": "0X2GJPNPFN",
  "uri": "m2://0X2GJPNPFN",
  "url": "https://math2.tamnd.com/wiki/halls-theorem-for-bipartite-graphs/",
  "writing": {
    "tag": "0X2GJPNPFN",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}